Calabi-Yau 4-folds and toric fibrations

TL;DR

The paper addresses identifying and enumerating fibrations in toric Calabi–Yau fourfold hypersurfaces by linking weight data to reflexive polyhedra and lattice projections. It develops a scheme to detect fibrations via subpolytopes and projections, and validates Hodge-number data using $Batyrev$'s toric formulas and $Vafa$'s Ramond-state counts. It shows transversality implies reflexivity for $n\le 4$ and reports that only about 20% of transversal five-dimensional polyhedra are reflexive; it enumerates $d\le150$ reflexive weights and $d\le4000$ transversal weights. The work provides a large, publicly accessible resource of weight systems and demonstrates that combining reflexive polytopes and weight data can yield roughly $3$ million elliptic fibrations, with important implications for F-theory model building and toric Calabi–Yau classifications.

Abstract

We present a general scheme for identifying fibrations in the framework of toric geometry and provide a large list of weights for Calabi--Yau 4-folds. We find 914,164 weights with degree $d\le150$ whose maximal Newton polyhedra are reflexive and 525,572 weights with degree $d\le4000$ that give rise to weighted projective spaces such that the polynomial defining a hypersurface of trivial canonical class is transversal. We compute all Hodge numbers, using Batyrev's formulas (derived by toric methods) for the first and Vafa's fomulas (obtained by counting of Ramond ground states in N=2 LG models) for the latter class, checking their consistency for the 109,308 weights in the overlap. Fibrations of k-folds, including the elliptic case, manifest themselves in the N lattice in the following simple way: The polyhedron corresponding to the fiber is a subpolyhedron of that corresponding to the k-fold, whereas the fan determining the base is a linear projection of the fan corresponding to the k-fold.